Hydraulic fracture mechanics, by far the most popular well stimulation technique, is often plagued by the uncertainties in field parameters for accurate field implementations. For vertical wells, uncertainties in reservoir parameters, such as far-field stresses, may only affect the size of fractures and do not pose many problems otherwise with respect to the geometry of the resulting fracture. However, for inclined (or deviated) wells, additional problems are introduced that cause a significant difference in the geometry of the fracture, both in size and shape, from its designed course, even in the near-wellbore region. Hence, all estimates of fracture behavior and post-fracture production should be made with the knowledge of the highly irregular fracture profile. More often than not, this is not done, causing considerable departures between expectations and reality.
The near-well stress concentration is affected by a number of factors, which include the far field stresses, the well deviation from both the vertical and a plane of principal stress, and the well completion configuration. In effect, the fracture initiation and, consequently, the resulting fracture geometry are greatly influenced by this stress concentration. Incomplete knowledge of all of these factors causes problems during execution of hydraulic fracturing, such as elevated fracturing pressures and unintended screenouts, because of tortuosity, which adversely affects the post-treatment well performance with especially severe effects in high-permeability formations. The uncertainty in magnitude and orientation of far-field principal stresses causes many of the unexplained perturbations in near-wellbore fracture profiles.
The far-field stresses, which are caused by overburden and tectonic phenomena, are supplanted by a new set of stresses when a borehole is drilled. This near-wellbore in situ stress field, in the presence of an arbitrarily inclined borehole, is dictated by the equilibrium equations and depends on the far-field stresses. Stress values are directly related to the state of strains through constitutive equations (elastic, plastic, etc.). When a hydraulic fracture is created at a borehole, the fracture initiation point is important to the fracture propagation, which, in turn, depends on the state of stress around the well. As a result, the presence of the fracture in the formation now redistributes the stresses from their original values without the fracture. In principle, if all of the required reservoir data are known, then the exact fracture profile can be predicted. However, in reality, uncertainty frequently is associated with the reservoir parameters, such as the principal stress orientations and, especially, the magnitude of the intermediate stress. An important consequence is that the resulting fracture geometry will not match its design. More important, in high-permeability fracturing, there is a compelling need to align the well, perforations, and the fracture to prevent or reduce very detrimental tortuosity.
For an open-hole completion, the problem has been studied previously and reported in P. Valko and M. J. Economides, “Hydraulic Fracture Mechanics,” Wiley, West Sussex, 1995. There are predictive models to evaluate both the fracture initiation pressure and the near-well fracture tortuosity, given the far-field stresses and all the angles that can describe the well position and the fracture initiation point. However, when a fracture is introduced into the formation, no closed form analytical solution is available, and numerical models must be relied on to compute the induced stress profile. Typically, finite element models are used predominantly in such solid mechanics applications.
In many cases, hydraulic fracturing may be performed on a completed well having a casing and sheath. The choice of sheath material, such as foamed cement or neat cement, may affect the fracture geometry significantly due to its material properties. Also, the presence of multiple zones may have other influences in the near-well zone, such as on fracture initiation and fracturing pressure. During hydraulic fracturing of a cemented well, for example, internally pressurized wellbores cause the casing to expand, which induces a tensile stress in the surrounding continuous cement sheath. As a result, the fracture initiation is a function of the cement's tensile strength and the tensile stresses induced within the cement sheath. However, the effect of the far-field stresses should be included in the field, which is almost always asymmetrical in nature. In effect, both tensile and compressive stresses may act on portions of the cement sheath, thereby making some portions more vulnerable to fracture initiation.
As mentioned, finite element models predominate in such applications. However, finite element modeling can become inefficient and cumbersome for many classes of problems, including fracture mechanics. Finite element models are cumbersome when it comes to complex geometry, in terms of their size, reusability with minor changes, and resources required. An alternative approach, the boundary integral equation method (BIEM), was proposed in the 1950's for fluid flow applications, and applied in the late 1960's to mechanical analysis. See, for example, C. A. Brebbia, “The Boundary Element Method for Engineers,” Pentech Press, Plymouth, 1978. The boundary element method (BEM) emerged as a more generally applicable technique during the 1970's, and has been developed substantially in the following years. See, for example, J. Trevelyan, “Boundary Elements for Engineers—Theory and Applications,” Computational Mechanics Publications, Southampton and Boston, 1994. Boundary element techniques are far superior to finite element models, due to ease of use, accuracy, flexibility, and computational speed.
The boundary element method is a numerical technique for analyzing the response of engineering structures when subjected to some kind of “loading.” The main feature of BEM is that the governing equations are reduced to surface or boundary integrals only, with all volume integrals removed by mathematical manipulation. Because only surface integrals remain, only surface elements are needed to perform the required integration. So, the boundary elements needed for a 3D (three-dimensional) component are quadrilateral or triangular surface elements covering the surface area of the component. Even simpler, the boundary elements for 2D (two-dimensional) and axisymmetric problems are line segments tracing the outline of the component.
The simplicity of a BEM model means that much detail can be included without complicating the modeling process. In particular, cylindrical holes, such as petroleum wells, can be modeled very quickly, where there is no connection between a hole and the outer surface. Boundary elements also allow analysis of problems that would overwhelm finite element models with too many elements. The system matrix for boundary elements is often fully populated (i.e., dense) and non-symmetrical, but is of significantly smaller dimension than a banded finite element global stiffness matrix.
Because boundary elements are simply lines for 2D and axisymmetric problems, there needs to be a convention used for determining which side of an element is the free surface and which side is inside the material. It is most convenient to use the direction of definition of the element connectivity as the indicator of this orientation. Under this convention, as will be appreciated by those skilled in the art, if the direction of all elements in the model were reversed, we would be modeling the entire infinite universe surrounding a void shaped like the boundary element mesh. In petroleum well applications, these boundary elements are very useful since a few elements can model the problem very accurately where several thousand finite elements likely would be necessary.
The boundary elements are located only on the surface of the component, as are the nodes of the elements. This means that the locations at which the boundary element results are found are only on the surface of the component. It is possible to extract the results for any internal point(s) inside the material simply from the solution over the boundary. The results are not just found by extrapolation, but by using an accurate integral equation technique very similar to that used for the solutions over the boundary elements.
Boundary elements also allow us to define models consisting of a set of sub-models, or zones. Zones are boundary element models in their own right, being closed regions bounded by a set of elements. They share a common set of elements with the adjacent zones. These “interface” elements, which are completely within the material and not on the surface, form the connectivity between the various zones. This zone approach can be employed when a component consists of two or different materials, when components have high aspect ratio, when elements become close together across a narrow gap leading to inaccurate results, or when computational efficiency needs to be improved.
This boundary element method eliminates the necessity for nested iterative algorithms, which are unavoidable when domain integral methods, such as finite difference methods and finite element methods, are used. Using BEM, it is easier to change a model quickly to reflect design changes or to try different design options. The boundary element method is highly accurate, because it makes approximations only on the surface area of the component instead of throughout its entire volume.